One of the most useful applications of probability to the Earth’s surface concerns uncertainty about location. Suppose the location of a point has been measured using GPS, but inevitably the measurements are subject to uncertainty, in this case amounting to an average error of 5m in both the east-west and north-south directions. Standard methods exist for analyzing measurement error, based on the assumption, well justified by theory, that errors in a measurement form a bell curve or Gaussian distribution. Spatially, one can think of the east-west and north-south bell curves as combining to form a bell. But the surface formed by the bell is not a surface of probability in the sense of Section 2.3.1, Spatial probability — it does not vary between 0 and 1, and it does not give the marginal probability of the presence of the point. Instead, the bell is a surface of probability density, and the probability that the point lies within any defined area is equal to the volume of the bell’s surface over that area. The volume of the entire bell is exactly 1, reflecting the fact that the point is certain to lie somewhere.
It is easy to confuse probability density with spatial probability, since both are fields. But they have very different purposes and contexts. Probability density is most often encountered in analyses of positional uncertainty, including uncertainty over the locations of points and lines.